Obama won the Nobel Peace Prize, but would receiving it be unconstitutional? This is the same legal issue that came up with former Senator Edward Kennedy's Knighthood.
The potential issue comes from the Constitution, Article I, Section 9, Clause 8 which reads; "No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State." .
Eugene Volokh explains the issue; "The Prize is apparently awarded by a Committee selected by the Norwegian Parliament; does it qualify as a present given by a “foreign State,” or does the Committee’s speaking for itself rather than for Norway suggest that the present does not come from a foreign State?"
As with the last time this issue came up, out of an abundance of caution Obama should ask for Congressional approval. Such a request and subsequent vote would not fail. Even if he does not, I do not see this going anywhere; but it is an interesting legal question.